Angle of Twist and Peak Torsional Shearing Stress

[r661559]

Homework Statement

A stepped solid shaft of circular cross section is rigidly clamped at ends A and D and loaded by twisting moments T1= 1kN.m and T2 = 1.5kN.m at points B and C. The material is steel for which G = 84x10^9 N/m^2, the length L of the shaft is 500mm, and the diameters of AB and CD are both 30 mm while that of BC is 50 mm. Determine the angle of twist at C and the peak torsional shearing stress.

http://images.4chan.org/sci/src/1362606506659.png

Homework Equations

Angle of twist θ= TL/JG
max torsional shearing stress τmax = Tc/J
centroidal moment of inertia J = (∏c^4)/2

The Attempt at a Solution

I have not attempted a solution because I do not know how to take into account that the shaft is rigidly clamped at both ends and what side to start from. Also, I do not know how the Torque at B (T1) comes into play with the angle of twist at c, I would imagine it increases the angle because it is in the same direction of Torque at C (T2)

 

[SteamKing]

You have the start of a FBD. Use the equations of statics to find the reactions required at A and D to keep the shaft in equilibrium.

 

[r661559]

Ok i understand by using the sum of the moment across the x plane Ta + Td = 2.5 kN.m. What's confusing me is if i break it into separate sections AB, BC and CD, or just AB and CD to find the reactions at A and D.

also, the total angle of the shaft is zero since both ends are restrained... so the angle of AB + angle of BC + angle of CD = 0 going from left to right angle of CD is negative.. I don't get where to add in the torque at B while finding the angle from b to c.

 

[SteamKing]

The shaft is statically indeterminate. Have you studied how to find the unknown reactions in such a case?

 

[r661559]

I must correct myself, I have seen how to solve for statically indeterminate shafts but it was a single cylinder with a single torque applied to the center. The two torques is confusing me.

when i try to make my second static equation, I am not sure how to take into account both torques applied.

 

[SteamKing]

Here is a similar problem, showing how to overcome the static indeterminacy:

http://www.mathalino.com/reviewer/m...-of-materials/solution-to-problem-319-torsion

 

[r661559]

thanks steamking, I think I've figured it out now.